Probing entropy bounds with scalar field spacetimes

We study covariant entropy bounds in dynamical spacetimes with naked singularities. Specifically we study a spherically symmetric massless scalar field solution. The solution is an inhomogeneous cosmology with an initial spacelike singularity, and a naked timelike singularity at the origin. We construct the entropy flux 4-vector for the scalar field, and show by explicit computation that the generalized covariant bound S_L(B,B^')<~[A(B)-A(B^')]/4 is violated for light sheets L(B,B^') in the neighborhood of the (evolving) apparent horizon. We find no violations of the Bousso bound [for which A(B^')=0], even though certain sufficient conditions for this bound do not hold. This result therefore shows that these conditions are not necessary.